A search for supersymmetry in events with large missing transverse momentum, jets, and at least one hadronically decaying tau lepton has been performed using 3.2 fb-1 of proton–proton collision data at s=13TeV recorded by the ATLAS detector at the Large Hadron Collider in 2015. Two exclusive final states are considered, with either exactly one or at least two tau leptons. No excess over the Standard Model prediction is observed in the data. Results are interpreted in the context of gauge-mediated supersymmetry breaking and a simplified model of gluino pair production with tau-rich cascade decays, substantially improving on previous limits. In the GMSB model considered, supersymmetry-breaking scale (Λ) values below 92TeV are excluded at the 95% confidence level, corresponding to gluino masses below 2000GeV. For large values of tan β, values of Λ up to 107TeV and gluino masses up to 2300GeV are excluded. In the simplified model, gluino masses are excluded up to 1570GeV for neutralino masses around 100GeV. Neutralino masses below 700GeV are excluded for all gluino masses between 800 and 1500GeV, while the strongest exclusion of 750GeV is achieved for gluino masses around 1450GeV.